Hyperinvariant subspaces for quasinilpotent operators on Hilbert spaces

In this paper, we employ the model theory due to C. Foias and C. Pearcy and the notion of Enflo's extremal vectors of quasinilpotent operators to study the hyperinvariant subspace problem for quasinilpotent operators. Our main result is that if a quasinilpotent quasiaffinity T has a sequence of “c-eigenvectors” xn of TnT∗n such that the set is compact, then T has a nontrivial hyperinvariant subspace.